fa.functional analysis - Boundedness of nonlinear continuous functionals

Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful.

The new version is about non-linear real-valued continuous functions on
$ell_infty(Gamma)$ where $Gamma$ has the cardinality of the continuum.
This can be slightly generalized as follows:

Let $kappa$ be an infinite cardinal and set $K$ to be
the closed unit ball of $ell_infty(kappa)$. Let
$f:Ktomathbb{R}$ be a continuous map. Does there exist
an infinite-dimensional subspace $E$ of $ell_infty(kappa)$
such that $f(Kcap E)$ is bounded?

If $kappa=aleph_0$, then a counterexample can be constructed.

On the other hand, if $kappa$ is a measurable cardinal, then
there exists a subspace $E$ of $ell_infty(kappa)$ which is isomorphic
to $c_0(kappa)$ and such that $f(Kcap E)$ is bounded. The argument
goes back to Ketonen. Let $FIN(kappa)$ be the set of all non-empty finite
subsets of $kappa$ and define a coloring $c:FIN(kappa)tomathbb{N}$
as follows. Let $c(F)$ be $n$ if $n$ is the least integer $m$ such that

$ max{ |f(x)|: xin span{e_t: tin F} and xin K } leq m $

where $e_t$ is the dirac function at $t$. Notice that $c$ is well-defined.
There exist $n_0inmathbb{N}$ and a subset $A$ of $kappa$ with $|A|=kappa$
and such that $c$ is constant on $FIN(A)$ and equal to $n_0$. If we set $E$ to be
the closed linear span of ${e_t: tin A}$, then $E$ is isomorphic to
$c_0(kappa)$ and $F(Kcap E)$ is in the interval $[-n_0, n_0]$.

Concerning the continuum: it might be that there are set-theoretic issues.
Firstly, let me recall that it is consistent that the the continuum is real-valued
measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy
(and quite advanced) machinery for ``killing" various Ramsey properties on
$omega_1$ (largely due to S. Todorcevic).

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A quick remark: there exists a non-linear continuous map $f:Ktomathbb{R}$,
where $K$ is the closed unit ball of $c_0(kappa)$ and $kappa$ is the continuum,
such that for every infinite-dimensional subspace $E$ of $c_0(kappa)$
the set $f(Kcap E)$ is unbounded.